According to a note posted on the Internet Movie Database House MD Forum, by rachel_lb "The theme song is 'Teardrop' by the group Massive Attack." and jupiter0225 posted that it was on the album Mezzanine. All of this is supported by the closed captioning during the credits during each episode.
***This album is also available from Amazon UK in Britain. But the overseas theme music is different. We can only assume that the contract for the use of this music was restricted to the U.S. So overseas there is theme music used for the first season and different theme music used for the second and that was done by the music supervisors and isn't available anywhere.

What was the music playing at the end of the Season Finale (as well as in the pilot)?

The music playing at the end of the season 1 finale "Honeymoon" was "You Can't Always Get What You Want" by the Rolling Stones. This is available on two albums: "Forty Licks" and "Let It Bleed"You can listen to a sample of this song at Amazon's pages on these albums (click on the album title above and again at the first Amazon page and go to the bottom of the next page to get a choice of which song to listen to). This song was also played during the pilot episode.